Strain and Vorticity Analysis Using Small-Scale Faults and Associated Drag Folds

Strain and vorticity analysis using small-scale faults and associated drag folds

Enrique Gomez-Rivas a,* , Paul D. Bons b, Albert Griera c, Jordi Carreras a, Elena Druguet a and Lynn Evans d

aDepartament de Geologia, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain bMineralogie und Geodynamik, Institut für Geowissenschaften, Eberhard Karls Universität, Sigwartstr. 10, 72076 Tübingen, Germany cLaboratoire des Mécanismes et Transferts en Géologie, Université Paul-Sabatier, 14 Ave Edouard Belin, 31400 Toulouse cedex, France dSchool of Earth Sciences, University of Melbourne, Victoria, 3010, Australia * Corresponding author. Tel: +34-935811163; Fax: +34-935811263; E-mail address: [email protected]

Published in Journal of Structural Geology, 29, 1882-1899. doi:10.1016/j.jsg.2007.09.001

This is the author’s version of the manuscript. You can access the copy-edited version at: http://www.sciencedirect.com/science/article/pii/S0191814107001642

Abstract

Small-scale faults with associated drag folds in brittle-ductile rocks can retain detailed information on the kinematics and amount of deformation the host rock experienced. Measured fault orientation ( α), drag angle ( β) and the ratio of the thickness of deflected layers at the fault ( L) and further away ( T) can be compared with α, β and L/T values that are calculated with a simple analytical model. Using graphs or a numerical best-fit routine, one can then determine the kinematic vorticity number and initial fault orientation that best fits the data. The proposed method was successfully tested on both analogue experiments and numerical simulations with BASIL. Using this method, a kinematic vorticity number of one (dextral simple shear) and a minimum finite strain of 2.5 to 3.8 was obtained for a population of antithetic faults with associated drag folds in a case study area at Mas Rabassers de Dalt on Cap de Creus in the Variscan of the easternmost Pyrenees, Spain.

Keywords: strain analysis; vorticity; progressive deformation; faults; drag folds; foliation.

1. Introduction associated drag folds in mostly ductile rocks (Passchier, 2001; Grasemann and Stüwe, One of the aims of structural geology is to 2001; Grasemann et al., 2003; Exner et al., determine and quantify the amount and type of 2004; Grasemann et al., 2005; Wiesmayr and deformation that rocks experienced. For this Grasemann, 2005; Coelho at al., 2005; Kocher structural geologist use a variety of structures and Mancktelow, 2006). In the modern that record deformation, such as folds, literature, these structures were first described boudins, veins, etc. (e.g. Ramsay and Huber, by Gayer et al. (1978) and Hudleston (1989). 1983). In this paper we propose a new method These structures were later dubbed "flanking to determine finite strain and the kinematics of folds" or "flanking structures" by Passchier deformation using isolated, discrete small- (2001), who used this term for a variety of scale faults and their associated drag folds. structures apart from fault-related drag folds. Slip along a fault will cause Instead of this new terminology, we prefer to heterogeneous deformation in the vicinity of use well-known and long-used terms: faults the fault. Drag folds are the usual result in and drag folds. foliated rocks. Recently, much attention has The aforementioned authors described been given to small-scale faults and their a range of drag fold structures and proposed a Gomez-Rivas, E., Bons, P.D., Griera, A., Carreras, J., Druguet, E., and Evans, L. (2007). Strain and vorticity analysis using small-scale faults and associated drag folds. Journal of Structural Geology, 29, 1882-1899 number of classification schemes. Basically, direction may change at a high strain and an isolated fault with its associated drag folds reverse drag folds then become normal drag falls into one of four categories by the folds. combination of two parameters: fault An isolated, discrete fault will typically movement is antithetic (a-type of Grasemann develop drag folds with a constant sign of et al., 2003) or synthetic (s-type) with regard curvature. In a ductile shear band (i.e. minor to the far-field sense of shear, and drag folds shear zone) the foliation is not cut by a fault, are normal or reverse with regard to the slip but can be traced continuously through the along the fault. Of these four, the antithetic shear band (Fig. 1b). This implies that there is reverse-drag category is the most common for an inflexion point where curvature changes isolated faults. The fact that reverse drag is sign (Coelho et al., 2005). This produces common is to be expected for isolated faults in shear-band type structures in the terminology an otherwise homogeneously deforming of Wiesmayr and Grasemann (2005). medium. A straight foliation element However, these structures still exhibit the (layering, cleavage) that is cut by the fault will same reverse or normal drag on a larger scale remain on a single straight plane away from than the deflection caused by the localised the fault, whereas close to the fault, it is bent shearing within the shear band. These reverse- by the fault movement (Fig. 1a). Both drag folds can usually not be discerned when synthetic and antithetic faults will therefore shear band spacing is on the same scale as the initially develop reverse-drag folds. However, reverse drag folds. Exner et al. (2004) showed that the slip

Fig. 1. Schematic illustration of the formation of reverse-drag folds adjacent to (a) isolated faults and (b) shear bands in a general shear field ( Wk =0.64) that is homogeneous far away from the fault/shear band. In case of a ductile shear band, normal drag is found within the shear band, in addition to reverse drag away from the shear band. The same applies to antithetic movement (top) and to synthetic movement (bottom). Left column shows the geometry before deformation.

Despite the several field studies (Gayer Koyi and Skelton, 2001; Harris et al., 2002; et al., 1978; Druguet et al., 1997; Harris, Exner et al., 2004; Kocher and Mancktelow, 2003), as well as numerical (Grasemann and 2006) of drag fold structures, so far only Stüwe, 2001; Grasemann et al., 2003; Kocher and Mancktelow (2005) proposed a Grasemann et al., 2005; Wiesmayer and way to use these structures to quantify the Grasemann, 2005; Coelho et al., 2005; Kocher finite strain and kinematics of deformation. and Mancktelow, 2006) and experimental They employed the analytical solution of simulations (Hudleston, 1989; Odonne, 1990; Schmid and Podladchikov (2003) for the

Gomez-Rivas, E., Bons, P.D., Griera, A., Carreras, J., Druguet, E., and Evans, L. (2007). Strain and vorticity analysis using small-scale faults and associated drag folds. Journal of Structural Geology, 29, 1882-1899 deformation field along an isolated fault that experienced multiple deformation phases itself is passively deformed by the applied during the Variscan Orogeny (Druguet, 1997; bulk flow. Their method is essentially Druguet, 2001; Bons et al., 2004). Some applying the reverse model strain field to quartzite beds, ranging from a few tens of straighten out the foliation. Since bulk centimetres to a few metres in thickness, are deformation kinematics and finite strain are intercalated in the meta-turbidites. They form not known a priori , a range of finite strains the only marker horizons that can be traced and vorticities are applied and the one that best over distances of up to a few hundred metres. straightens the foliation is chosen as the All the drag fold structures discussed in this solution. The advantage of the method is that a paper were found in one such bed, which has a single structure can be used to determine the distinct black-and-white cm-scale banding vorticity of deformation, the finite strain since (Fig. 2). The banding is layer-parallel and formation of the fault, and the original therefore assumed to be original sedimentary orientation of the fault relative to the foliation. layering. The colour difference is a result of A disadvantage is that appropriate software is different amounts of graphite and other needed. impurities, which also results in a difference in In this paper we propose a similar grain size between the layers (Fig. 2g-h). method to determine these three parameters. There are no indications for any significant The advantage of our proposed method is that differences in rheological properties between the method does not necessarily require a the dark and light bands (no cuspate-lobate computer. Instead, charts can be used, which structures, buckle folds in specific layers, etc.). means the method can easily be applied in the Near Mas Rabassers de Dalt, the field. However, a more accurate determination quartzite and S 1 layer-parallel foliation (S 01 ) can only be achieved numerically, as is are affected by two more folding events (D 2 described in this paper. A disadvantage is that and D 3), resulting in a complex exposure multiple fault-drag fold structures are needed pattern (Druguet, 1997). Pegmatites that at different stages of development (finite strain intruded during peak-metamorphic conditions since formation). This study is based on a (Druguet and Hutton, 1998) are only affected population of drag fold structures in deformed by retrograde D 3 folding and shearing. The quartzites from Mas Rabassers de Dalt on the regional trend of the S 01 main foliation is NW- Cap de Creus Peninsula in north-eastern Spain SE, when not affected by D 3 shearing. A broad (Fig. 2). These structures and their setting will zone of dextral NW-SE-trending D 3 shearing be described first to provide the background rotated S 01 towards the NE-SW trend that for the method that is described in the dominates in the area shown in figure 3. The subsequent sections. curvature of the quartzite layer and the S 01 foliation is due to decametric folds predating 2. Examples from the Rabassers quartzite the shearing event that can be recognized from the structural map. Localisation of the shearing 2.1. Regional setting of the Mas Rabassers de led to the formation of a number of narrower Dalt locality (≥10 m) shear zones with distinctly elevated shear strain. Although the superposition of The Cap de Creus Peninsula is the three deformation events makes it difficult to easternmost outcrop of the Variscan basement interpret the map pattern at Mas Rabassers de exposed along the Axial Zone of the Pyrenees Dalt (Fig. 3), extensive mapping at the site and (Barnolas and Chiron, 1996; Carreras, 2001). the region has unambiguously established the The dominant lithology in the area of interest dextral nature of the D 3 shearing and near the ruin of Mas Rabassers de Dalt (UTM associated folding (Carreras and Casas, 1987; 31N 0523100, 4685200, Fig. 3) is a Carreras, 2001; Carreras et al., 2005; Fusseis monotonous series of amphibolite-facies meta- et al., 2006). turbidites (Druguet, 1997; 2001). The rocks

Fig. 2. Drag fold structures in the banded quartzite at Mas Rabassers de Dalt, Cap de Creus, Spain. Sense of shear is top (east) to the right. (a-d) Antithetic faults with reverse-drag folds at different stages of development. (e) One of the rare synthetic faults. (f) Photograph and sketch showing that in the third dimension the faults are straight and extend further than their length perpendicular to the banding. (g) Plane-polarised light micrograph of an antithetic fault. Variations in the content of graphite and mica particles form the dark and light bands. (h) Same image in cross- polarised light. Quartz grain size is largest in clean quartz. All images looking onto the surface perpendicular to the foliation and faults. Black scale bars 10 mm, white scale bars 0.5 mm, Ø of 5 €-cent coin is 21 mm.

Fig. 3. Detailed map of the Mas Rabassers de Dalt Outcrop showing the refolded quartzite bed and the localities where the small-scale faults were found and measured. The stereoplot summarizes the main structural information: poles to quartzite bedding (open dots) that lie on great circle (dashed line) defining the D 3 fold axis (closed dot) which lies on the great circle of the average D 3 shear plane. The cross is the average D 3 shear direction. Black arrows indicate sense of D3 shear zones. (Based on Druguet, 1997).

2.2. Drag fold structures measured at the fault, preferably in the middle of the fault, and finally the ratio between the The discrete faults with drag folds are thickness of a marker layer at the fault, mainly found in the refolded quartzite bed. measured parallel to the fault ( L) at the fault, Within that bed they only occur in sections of and perpendicular to the layer ( T) away from the bed that run roughly parallel to the D 3 the fault. All parameters must be measured in shear zones. This suggests that they formed the plane perpendicular to the fault and during D 3 dextral shearing and not during foliation. earlier deformation events. Almost all The first main assumption is that the (localities A-F in Fig. 3) occur in the western fault acts as a passive, straight marker line that limb of a D 2-fold, of which the hinge is skirted is being rotated and stretched/shortened by the by one of the zones of most intense localised applied bulk flow. This assumption is shearing. validated by both numerical and physical The drag folds form at cm-scale, experiments (Grasemann and Stüwe, 2001; steeply plunging faults, best seen on gently Exner et al., 2004). Clearly, the four dipping outcrop surfaces. Almost all drag folds parameters will evolve from their initial values are reverse. Faults with the least offset relative (α0=β0 and L 0/T 0=1/sin( α0)), depending on the to their length are almost perpendicular to the flow field relative to the initial orientation of banding in the quartzite (Fig 2a). More the fault and foliation. We need to know how evolved structures make an increasingly α, β and L/T evolve, as a function of smaller angle with the banding, suggesting the progressive deformation and initial conditions, structures progressively rotated clockwise to determine which initial conditions, (Fig. 2c-d). Fault tips can rarely be discerned, kinematics of flow and finite strain lead to the as faults tend to bend in a listric form to combinations of α, β and L/T that were become parallel to the banding at both ends. measured in the field. However, there may not Dextral layer-parallel slip is observed in a few be a unique solution for any given single rare cases where crosscutting veins are offset. combination of α, β and L/T. This brings us to Clockwise rotation of the faults and layer- the second main assumption for the proposed parallel slip all indicate dextral shear. The method: during progressive deformation faults faults are therefore interpreted as antithetic form at different stages, but with the same faults. Synthetic faults (Fig. 2e) are rare in the initial orientation ( α0). At the end of quartzite, and usually make a small angle with deformation (the state observed in the field), the banding. In the third dimension, the small each fault experienced different amounts of faults are remarkably straight and may extend deformation and is therefore in a different state over more than a metre (Fig. 2f). Even on the of development (Fig. 2a-d). Analysis of microscopic scale, the faults are discrete several of such faults produces a number of planes with only a very narrow damage zone different α, β and L/T combinations. These (Fig. 2g-h). measured combinations should lie on a path in

α, β and L/T space that is unique to the flow 3. Method kinematics and initial orientation of the faults.

The basic idea of our proposed method The method described below is aimed at is that theoretical paths for all flow kinematics estimating the kinematic vorticity number and initial fault orientations can be (Means et al., 1980) and finite strain that the determined, and can then be compared with α, rock experienced using parameters of the drag β fold structures that can be measured easily. and L/T data sets that are measured in the The following parameters can be determined field. The path that best fits the data provides us with the flow kinematics and the initial fault in the field (Fig. 4): the angle ( α) between the orientation. It also allows us to determine fault and the far-field foliation, the drag angle which data point represents the highest strain, (β) between the foliation and the fault which gives a minimum estimate of the finite 6

Gomez-Rivas, E., Bons, P.D., Griera, A., Carreras, J., Druguet, E., and Evans, L. (2007). Strain and vorticity analysis using small-scale faults and associated drag folds. Journal of Structural Geology, 29, 1882-1899 strain. Comparison of theoretical paths and Ebner and Grasemann, 2006). As in most data can be done using charts or with a studies, we assume that the kinematics of computer program that carries out the best fit. strain do not change during deformation. The The advantage of using charts is that they can bulk flow field is now given by the position easily be employed in the field. gradient tensor F:  a g  F = , (1)  0 1/ a where a is the amount of stretching, and g the amount of shearing, both parallel to the foliation. Because of the definition of the reference frame, the far field foliation does not rotate relative to the reference frame, but it may stretch or shorten if a≠1. F is area- conservative because of our assumption of plane-strain flow. It is also assumed that the fault is frictionless, so that it cannot support any shear stress. This implies that the material adjacent to the fault stretches/shortens in pure shear parallel to the fault. Rotation of the fault adds

a spin to the deformation, but deformation Fig. 4. Sketch showing a fault and drag fold in the undeformed (a) and deformed (b) stage, with all the immediately adjacent to the fault plane parameters that are required for the analysis. remains coaxial. The no-friction assumption is unlikely to be completely valid in reality. 3.1. Theoretical α-β-L/T paths However, our experiments and numerical simulations below show that small deviations The following analysis is based on the in mechanical properties of the fault or shear deformation at an isolated single straight fault zone do not noticeably change the outcome. in an otherwise homogenously deforming Furthermore, the fault as a whole behaves as a medium. The fault is supposed to have a passive plane, or a line in 2D, and therefore limited extent, so that the offset reduces to stretches and shortens according to the bulk zero at both ends. We consider a plane-strain flow field. We define e as the amount of case, with the fault oriented parallel to the stretch or longitudinal strain of the fault (its intermediate principal stretching direction. The finite length / original length). With these problem can therefore be regarded as two- assumptions an analytical solution exists for dimensional. If deformation is not plane strain, the evolution of α, β and L/T for a layer that stretching or shortening in the third dimension intersects the fault at its centre. would cause an area change in the section To determine the orientation of the under consideration, but no changes in the fault with progressive strain, we consider a angles and other parameters that are used unit vector parallel to the fault. This vector has below. We further consider an initially straight initial coordinates [cos( α0),sin( α0)]. After foliation perpendicular to the section under deformation, and due to the application of the consideration. tensor F, the vector will have new coordinates Similar to Kocher and Mancktelow [a·cos( α0)+ g·sin( α0),(1/ a)·sin( α0)]. The stretch (2005), we fix our reference frame to be (e), parallel to the fault, is the ratio of the finite parallel and perpendicular to the far-field and original length of the unit vector: foliation orientation. The foliation is assumed to be parallel to a direction of zero rotation and 2 1 e= ()a ⋅cos ()α + g⋅sin ()α + sin 2()α (2) therefore parallel to one of the flow 0 0 a2 0 eigenvectors or apophyses (Passchier, 1988; 7

This line L0 gets stretched by the same amount The finite orientation ( α) of the fault (e) as the fault, so its length after deformation relative to foliation is: will be: e⋅T L = e⋅ L = 0 (8)   0 α sin (α ) sin ()0 α = arctan 0  (3)  2 ⋅ α + ⋅ ⋅ α   a cos ()0 a g sin ()0  As the absolute dimensions are irrelevant for As deformation progresses the foliation the geometry of the system, we combine is reoriented at the fault, describing a drag equation (7) and (8) to obtain the ratio L/T : angle ( β) between the foliation and the fault e⋅T a e⋅ a plane. We use the assumption of a frictionless L /T = 0 = (9) ()α T ()α fault and therefore pure shear parallel to the sin 0 0 sin 0 fault. The foliation at the fault thus experiences a stretch ( e) parallel to the fault, We now have the three measurable while it passively rotates along with the fault. parameters α, β and L/T as a function of the Stretching and rotation determine the drag unknown variables α0, a, and g. Although the angle. A local position gradient ( F’ ) tensor can combination of a and g defines the amount of be defined in a coordinate system parallel to finite strain and the kinematics of strain, it the fault: may be more useful to use the two variables finite strain ration ( Rf) and angle between the  e 0  two flow apophyses on the vorticity-normal F'=   (4)  0 1/e section (ω) or kinematic vorticity number (Wk ). Rf is the axial ratio of the finite strain ω A unit vector in this local coordinate ellipse and the angle between the flow system will change from initial coordinates apophyses, with: β β 1/ arc tan aa gω −=  [cos( 0),sin( 0)] to new coordinates ()cos Wk ω= β β β [e·cos( 0),(1/ e)·sin( 0)]. The angle ( ) between , (10) foliation at the fault and that fault will then be (using α0 = β0): and

    (β ) ( ) 2 2 2 2 sin 0 sin a0 + 1 ()+ + + ()− β =   =   2 g 4 1/ a a g a 1/ a arctan 2  arctan 2  (5) = ⋅ β ⋅ Rf (11)  e cos ()0   e cos ()a0  2 2 2 g2 + 1 ()1/ a + a − g2 + ()a −1/ a 4 The reference layer should intersect the fault just at the centre of it, where the ω can range from 0° for simple shear ( Wk =1) maximum displacement can be found. Away to +90° for pure shear ( Wk =0) stretching from the fault the finite thickness ( T) of that parallel to the foliation, or -90° for pure shear layer is a function of the bulk finite strain and shortening parallel to the foliation. α its original thickness ( T0): With the above equations, curves of and L/T as a function of β are shown for T different vorticities and starting orientation of T = 0 (6) a the fault (Fig. 5).

The initial fault-parallel thickness ( L0) is: 3.2. Determining vorticity and initial fault angle with charts T L = 0 (7) 0 α ω sin ()0 To determine the vorticity ( ) and initial fault α α β orientation ( 0), , and L/T need to be

Gomez-Rivas, E., Bons, P.D., Griera, A., Carreras, J., Druguet, E., and Evans, L. (2007). Strain and vorticity analysis using small-scale faults and associated drag folds. Journal of Structural Geology, 29, 1882-1899 measured on a population of faults with drag folds. Such data can be measured in the field 3.3. Numerical implementation of the method or from pictures. The plane of observation should be perpendicular to both fault and Finding the curve that best fits the data can foliation. If not, the apparent values should be also be done numerically, using a least-squares corrected to get the true values. The data approach. A small program that does the curve should be collected as close as possible to the fitting was written in the language "C" (source middle of the fault, so that equation (2) holds code can be obtained from the authors). Input for the stretching of the material immediately is a text file containing a list of α, β and L/T adjacent to the fault. It should also be noted data. The program cycles through all possible that these equations can be only used when the kinematic vorticity numbers (-1 to +1) and α0 fault is discrete. In case of a narrow ductile angles (0 to 180°), each with increments of 1°. β shear band, the angle would be modified due For each ω and α0 combination, the program to shearing of the foliation in the narrow zone then calculates the α-β-L/T curve for (Fig. 1b). progressive strain, increasing strain in small Applying the above equations, several increments. For each strain increment and each α β unique graphs for the evolution of ’, and i-th data point, the difference ∆i between the L/T can be plotted for progressive strain, for a theoretical and measured α, β, L/T values is certain starting orientation of the fault ( α0) and calculated: a certain kinematic vorticity number ( Wk ) (Fig ()()αα−2 + ββ − 2 5). Charts covering the full range of ω from - ∆ = ci ci i(α ,, ω Rf ) (12) 90 to +90° and α from 0 to 180° are provided 0 +() − 2 0 wLT/c LT / i in the appendix. We assume that all the shear bands start off at different times, but with a Here the subscript c stands for similar orientation. Each fault then represents theoretical values and i for measured data. a different stage of development. The data can Because the range of L/T values differs from be plotted in each of the graphs of figure 5. that of the angles α and β, L/T data may be Ideally, all data should plot on a single curve given a different weighting ( w) for the least- that represents the evolution of a fault system squares best fit. For a given ω and α with a certain α and ω. The fault that 0 0 Σ∆ ∆ experienced the least strain should lie closest combination, the sum ( i) of the smallest i- α value for each data point is a measure of how to the estimated initial fault angle ( 0). The ω α total amount of strain can be estimated from well that and 0 combination fits the data. ω α Σ that for the most developed fault system. This The and 0 combination with the lowest ∆i ω α is, of course, a minimum estimate, because is regarded as the best estimate of and 0. ω α even the most developed, oldest fault that is Once a best estimate for and 0 is found, found must not necessarily have experienced one can estimate the amount of strain that each the total finite strain of the host rock. The pair analysed fault experienced by finding the of curves in figure 5 that best fits the eight strain that minimises ∆i, using equation (12). measurements is the one for ω=0° (simple shear) and α0 is 70º to 80°. The highest strain the rock experienced is estimated to be between Rf=8 and 16, which corresponds to a dextral shear strain of 2.5 to 3.8. Figure 5 shows that the curves for different α0 and ω are distinct, as long as α0 is larger than about 40°. This means that the method is only applicable to faults that started off at a high angle to the foliation.

Fig. 5. Curves of α and L/T as a function of β for different vorticities and starting orientations of the fault. This chart can be used to estimate Rf , vorticity and initial fault angle in the field. Insets show the Mohr-circle for stretch for an Rf-value of 4. Eight data points from Rabassers de Dalt are plotted in each of the graphs. The pair of curves that best fits these data is the one for ω=0° (simple shear) and α0 is between 70º and 80°. The highest strain the rock experienced is estimated to be between Rf=8 to 16 (shear strain is 2.5 to 3.8).

4. Validation of the method ω=3° or Wk =1.00 (true value 0° and 1.00 respectively). 4.1. Introduction

In order to ascertain the validity of the method, it has been tested on several analogue and numerical experiments with different initial fault angles and different boundary conditions. First, the method has been applied to a simple shear analogue model from Exner et al. (2004) and later to a pure shear experiment of our own. We also ran a series of numerical experiments with a variety of initial angles and vorticities, ranging from pure to simple shear. In all cases, we measured α, β and L/T of a single drag fold structure at different stages of its development, and applied the least-squares best-fit routine to the data.

4.2. Validation on a simple shear analogue experiment

Exner et al. (2004) studied drag fold structures at a fault in a deforming a homogeneous, linear viscous matrix material (PDMS) in a ring shear rig. Each of their Fig. 6. Progressive development of a reverse a-type models started with a predefined fault, flanking fold (Modified from Exner et al. (2004). lubricated using liquid soap and silicone oil. They tracked the offset and deflection of 4.3. Validation on pure shear analogue foliation around the fault using a marker grid. experiments We used published images of one experiment To test the method on a pure shear case, we for α =90° according to the authors (Fig 6, 0 used the deformation apparatus described by after their figure 6). It should be noted that the Carreras and Ortuño (1990) and Druguet and actual starting angle in that experiment was Carreras (2006). The deforming medium was slightly less, about 87°. The fault initially has soft, commercially available plasticine. This antithetic slip and develops reverse drag folds. material has been characterized as non-linear At a shear strain of 2.3, the fault has rotated elasto-viscous with a stress exponent of 3, an 67° and slip reverses to become synthetic. In η∼ 7 the terminology of Grasemann et al. (2003) the effective viscosity 4·10 Pa·s at the ρ system evolves from a reverse-drag a-type, to experimental conditions, a density of 3 3 ∼ 5 a normal-drag s-type flanking fold. 1.15·10 kg/m , and shear modulus G 10 Pa Nine groups of data ( α, β, L/T ) were (Gomez-Rivas, 2005). The model had initial measured from the figures of Exner et al. dimensions of 29x15x10 cm and was deformed in pure shear at a temperature of (2004) up to a shear strain of 1.8, where the -5 -1 finite offset along the fault is still antithetic. 26ºC and at a strain rate of 4·10 s . Since With our analysis (Fig. 7) we obtained an tests with a lubricated cut, as used by Exner et estimated initial fault angle of 85° (true value al. (2004), failed, we simulated the fault with a 87°) and an angle between flow apophyses of lenticular fracture that was filled with much softer PDMS (Fig. 8). The fault was initially

Gomez-Rivas, E., Bons, P.D., Griera, A., Carreras, J., Druguet, E., and Evans, L. (2007). Strain and vorticity analysis using small-scale faults and associated drag folds. Journal of Structural Geology, 29, 1882-1899 oriented 60° to the extension direction. A 5 was applied by moving the sides of the mm grid was drawn on the surface of the sample, while keeping the sample thickness plasticine. Plane-strain pure-shear deformation constant at 10 cm.

Fig. 7. Curves of α, β and L/T for simple shear and a starting orientation of the fault of 85º, which best fit the data measured from the experiment of Exner et al. (2004).

Shortening lead to a rotation of the 4.4. Validation on finite element numerical fault and the development of reverse-drag simulations folds. The soft PDMS was squeezed towards the tips of the fault, where wing cracks As shown above, our proposed method developed. Despite these developments, our appears to work well for ideal pure and simple analysis of six data groups, measured every shear deformation. Unfortunately, 10% shortening, gave a good estimate of the experimental data were not available for vorticity (89° instead of 90°) and initial fault general shear. We therefore conducted a series angle (59° instead of 60°) (Fig. 9). of finite element models to test the method for a range of vorticities and initial fault angles. For the numerical simulations we used the code BASIL (Barr and Houseman, 1996) that is linked to the modelling platform Elle (Jessell et al., 2001). The models were two-dimensional and consisted of a square containing a narrow ellipse in the centre (Fig. 10). The host rock was simulated with a homogeneous isotropic linear viscous material with a viscosity ( η) of one. A single layer of viscosity 1.1 represented the foliation in the host rock. Like Grasemann et al. (2003) we simulated the fault in the centre of each model with a narrow ellipse with a viscosity of 0.01. Two types of initial geometries were considered, with the ellipse oriented at 45º and 75º to the foliation, respectively. These two models were deformed under different velocity boundary conditions, from simple to pure shear, varying the angle between flow apophyses ( ω) by 30º (Table 1). Fig. 8. Initial and final stage of a pure shear analogue The grid was generated with a self-meshing model showing the evolution of an pre-existing fault (a PDMS-filled lens) and its associated drag folds. The routine using Delauney triangles with a side of each square is 0,5 cm. wide. minimum angle of 10º. 12

At least 6 groups of data ( α, β, L/T) estimated initial fault angles are within 7º of were measured at different finite strain from the true values. each simulation, and analyzed to determine the Summarizing, in all tests the results vorticity and initial fault angle. The difference from the analysis closely match the known between true and estimated values are plotted true values of the physical and numerical in Fig. 11, and listed in table 1. Differences in experiments, allowing us to apply the method Wk ranges from 0 to 0.1 at the most, and to naturally deformed rocks with confidence.

Fig. 9. Curves of α, β and L/T for pure shear and a starting orientation of the fault of 59º. The measured data points of our experiment fit precisely to the calculated curves.

Fig. 10. (a) Initial configuration in finite element simulations with BASIL for an initial fault angle of 75º. (b) Geometry at the end of a simulation for Wk =0.5 at a finite strain of Rf =2.6.

5. Strain analysis applied to the Mas recorded by fault structure number 5 (at Rabassers de Dalt outcrop locality C in Fig. 3) with a finite strain of about Rf =8 to 16, which is equivalent to a A total of 29 small antithetic faults in the shear strain of about 2.5 to 3.8. quartzite layer at Mas Rabassers de Dalt (Fig. The dextral simple shear inferred from 3) were analyzed to estimate the deformation this analysis is consistent with the field experienced by this rock. The finite fault observations: the quartzite layer is oriented orientations ranged from α=10 to 64° (Table parallel to the zone of highest D3 shear strain. 2). The data were processed with the software Only one small fault (locality F) was found described in section 3.3. The results showed an away from the main shear zone, but this one, α initial fault angle ( α0) of 78º and an angle with a high angle of =54° to the foliation, between flow apophyses ( ω) of 3º, which gives experienced less finite shear strain. a bulk kinematic vorticity number ( Wk ) of 1.00 (dextral simple shear). The highest strain was 13

17, 21 and 25 data and using these subsets to determine vorticity and initial fault angle. Ten different random subsets were processed for each size of the subset. Figure 13 shows that even very small datasets (5 to 9 data) already give approximately the right solution. It should also be noted that least-squares best fit using 29 data points (measured by EGR) produced almost identical results to that using the graphs (Fig. 5) on only eight data points

Fig. 11. Comparison of true (open dots) and estimated independently collected by someone else (closed dots) values of kinematic vorticity number and (PDB). This not only indicates that the initial fault angles, for eight numerical simulations with graphical method with a limited data set BASIL. produces good results but also that user bias does not seem to be a significant factor in the Table 1 analysis. True and calculated values of bulk kinematic vorticity number ( Wk ) and initial fault angle ( α0) for eight finite element simulations, showing that errors in Wk are 6. Discussion and conclusions below 0.1 and in α0 below 7º

Initial ω Initial Calculated Wk Initial Calculated α0 In this paper we have shown that α α angle Wk Wk error 0 0 error small-scale faults with drag folds can be used 0º 1.00 0.92 0.08 45º 52.0º 7.0º to determine vorticity, initial fault angle, and 0º 1.00 1.00 0.00 75º 75.1º 0.1º 30º 0.87 0.79 0.08 45º 48.8º 3.8º estimate of the minimum finite strain since 30º 0.87 0.86 0.01 75º 69.0º 6.0º first fault nucleation. This is a useful addition 60º 0.50 0.48 0.02 45º 44.7º 0.3º to the structural geologist's "toolbox" because 60º 0.50 0.58 0.08 75º 72.7º 2.3º 90º 0.00 0.01 0.01 45º 44.1º 0.9º relatively few methods exist to determine and 90º 0.00 0.10 0.10 75º 75.7º 0.7º quantify vorticity (Ghosh, 1987; Passchier and Urai, 1988; Wallis, 1992; Short and Johnson, The available data set it is large enough 2006). to test the precision of the method. This was done by randomly selecting subsets of 5, 9, 13,

Fig. 12. Curves of α, β and L/T for an angle between flow apophyses of 3º and an initial fault angle of 78º. The plotted data correspond to the measured parameters at Mas Rabassers de Dalt.

The initial fault angle ( α0) can usually independently first, it can of course be used in be estimated in the field, by finding the the subsequent determination of the angle steepest fault with the least offset and drag between flow apophyses (ω), either when fold bending. If this α0 is determined using the graphs (Fig. 5 and Appendix), or

Gomez-Rivas, E., Bons, P.D., Griera, A., Carreras, J., Druguet, E., and Evans, L. (2007). Strain and vorticity analysis using small-scale faults and associated drag folds. Journal of Structural Geology, 29, 1882-1899 when using the least-squares technique. In the parallel to the shear plane. With the new latter case one can set α0 and only iterate over analysis of the faults with drag folds the ω and Rf to find the best fit. However, without kinematic vorticity number is better using a priori knowledge of α0, the method constrained. appears robust and produced estimated values close (<10°) to the true ones in all tests on experiments and numerical simulations.

Table 2 Values of α, β and L/T measured from the Rabassers de Dalt outcrop locality data group α0 β0 L/T A 1 14 39 2.93 2 36 67 1.93 B 3 53 77 1.56 C 4 21 28 2.81 5 10 31 2.98 6 28 61 1.89 7 13 21 3.31 Fig. 13. Graph showing the stability of this analytical D 8 30 39 1.83 method using a different number of groups of 9 26 53 1.50 measurements. The solution becomes stable using less 10 35 46 1.42 than 10 groups of data. 11 33 50 1.75 12 63 84 1.30 13 26 60 1.65 In conclusion, we propose a new 14 29 56 1.91 method to determine vorticity, initial fault 15 19 49 1.95 16 42 73 1.62 angle and finite strain using small-scale faults 17 54 66 1.28 with drag folds. Theory and validation tests on 18 39 69 1.83 experiments and numerical simulations show 19 64 81 0.94 20 34 61 1.50 that the method is robust, provided the 21 36 52 1.71 following assumptions hold: (a) the structures 22 43 62 1.50 nucleate at different stages during 23 21 54 2.13 E 24 27 41 1.71 deformation, and therefore record different 25 23 57 1.53 amounts of strain, (b) the faults all nucleate in 26 22 26 2.42 α 27 56 83 1.02 approximately the same orientation ( 0), (c) F 28 51 81 1.08 the flow kinematics do not change during G 29 54 76 1.15 deformation, (d) the structures are isolated to avoid interference between adjacent structures, In the field study with 29 measured and (e) the faults are discrete, so that the drag faults, simple shear deformation was obtained, angle ( β) can be determined accurately. The which is consistent with the known local last assumption means that ductile shear bands deformation at Mas Rabassers de Dalt. Still, (Fig. 1b) are not suitable for this method. one cannot determine the exact vorticity that Although a least-squares best fit the quartzite experienced with only routine is preferred to obtain the best estimate orientations of foliations, fold axes, and other of kinematic vorticity number, initial fault structural elements. Local field observations angle and minimum finite strain, charts can be made so far only indicated a dominant simple used to obtain a first estimate. shear component, leaving open sub-simple shear with some shortening or stretching

Fig. A1. Definition of parameters

Fig. A2. Plot your data on a transparency using these blank charts

Acknowledgements · The ratio ( L/T ) of the thickness of a layer away from the fault ( T) and the thickness of This work was financed through the PhD grant BES- the same layer parallel to the fault and at 2003-0755 to EGR and research project CGL2004- the fault ( L). 03657, both funded by the Spanish Ministry of Education and Science. We thank Jens Becker and Anne Each pair of graphs is for a certain Peschler for their help with the BASIL modelling. We vorticity, defined by the angle between the gratefully acknowledge D. Jiang and T. Bell, whose flow apophyses ( ω) or the kinematic vorticity constructive reviews greatly improved the manuscript. number ( Wk ). Arrows in the graphs show α-β (left) and L/T -β (right) paths as a function of Appendix A. Charts to estimate initial fault increasing finite strain and initial fault angle ( α ), vorticity ( ω) and minimum finite 0 orientation ( α0). Dashed lines are finite strain strain ( Rf ) contours at Rf = 2, 4, 8, and 16. Plot your measurements on a To use these charts, measure the following transparency, using the blank pair of graphs parameters from a number of fault systems, provided (Fig. A2). Then overlay your plot on preferably at different stages of development the graphs (Figs. A3 to A6) and find the (Fig. A1): vorticity where your data most closely follow · The angle between the fault and the one single arrow on both graphs. The arrow foliation away from the fault ( α); fitting your data points provides you with the · The angle between the fault and the starting orientation of the faults. Ideally, each deflected foliation at the fault ( β); data point should have the same finite strain in both graphs as well.

Fig. A3 to A6. Curves of α and L/T as a function of β for different vorticities and starting orientations of the fault

(α0). Insets show the Mohr-circle for stretch for an Rf-value of 4. Dashed lines are finite strain contours at Rf = 2, 4, 8, and 16.

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